BOSTON – While Speaker Robert DeLeo said Monday he hoped to pass transgender anti-discrimination legislation this session, the House of Representatives voted to give the committee reviewing the measure more time before it must make a recommendation.

DeLeo, who supports the bill that would provide anti-discrimination protections to transgender people using public accommodations, said he still does not know whether the House would have the votes to override a possible veto.

Gov. Charlie Baker said Monday that his position on the measure wasn’t altered by declarations of support from the public education community, from teacher unions to superintendents and school committees made Monday morning. The Massachusetts Teachers Association and the American Federation of Teachers, the two major Bay State education unions, joined with the Massachusetts Association of School Superintendents and the Massachusetts Association of School Committees to announce support of the transgender bill.

“We certainly support nondiscrimination in Massachusetts for anybody. But the details on this one are important, and I know the Legislature’s been working on it, and we look forward to seeing what they produce,” Baker told reporters after meeting with DeLeo, a Winthrop Democrat, and Senate President Stan Rosenberg (D-Amherst) – who both support the bill.

Its fate uncertain, legislation providing transgender people with new anti-discrimination rights might not join the flurry of bills moving out of committees ahead of a Wednesday deadline.

The bills (S 735/ H 1577) were referred to the Judiciary Committee in January. While Wednesday is the deadline for committees to report out timely filed legislation, those two bills are part of an order seeking to extend the deadline to May 2, according to a committee aide.

Baker on Monday said he isn’t sure whether he would veto the transgender bill should it reach his desk. “It would depend on what it says,” he said. Baker opposed a prior version of the bill.

The legislation would provide anti-discrimination protections to transgender people using public accommodations and would let them use public locker rooms and restrooms associated with their gender identity rather than their anatomical sex.

Opponents of the measure have focused on its effect on young people, raising the specter of males masquerading as transgender females to gain access to the girls’ bathrooms and locker rooms.

“I hope there will be severe consequences at the ballot boxes of anyone who chooses not to protect our children,” Rep. Marc Lombardo, a Billerica Republican, told the News Service earlier this year.

Proponents counter that schools already accommodate transgender students. After the 2011 law took effect in 2012, the Department of Elementary and Secondary Education advised that principals should clarify transgender students “may access the restroom, locker room, and changing facility that corresponds to the student’s gender identity.”

DeLeo has sought to gauge support among House Democrats for the bill, but said Monday that little has changed over time as lawmakers have had a chance to study the proposals.

“Those who are for it really have stayed the course and are still wanting to move forward. So it’s probably about in the same place as it was before,” said DeLeo, who said members are still relaying the “same concerns” about the legislation. Asked whether the House has the votes to override a veto, DeLeo said, “I’m not sure of that.”

DeLeo said the prospects of the bill’s passage become dimmer as the end of session in July draws nearer.

“My hope is really that we do it this session. The closer it gets to the end of the session it becomes, I suppose, more doubtful, but my hope still is that we try to get it done this session,” DeLeo said.

July 31 – a Sunday – is the last day of formal sessions when roll call votes can be taken and controversial bills can pass in the face of opposition from a minority of the members. The governor has 10 days to review legislation before either amending and sending it back, vetoing it or signing it into law.

Attorney General Maura Healey, a Democrat who has likened the issue to the civil rights struggles if the 1960s, will meet with transgender youth and their families on Tuesday. She plans to host discussion with them and supportive business leaders on the importance of “equality and public accommodations” for transgender people.